JP-10-73049A (U.S. Pat. No. 5,806,506) shows a cylinder-by-cylinder air-fuel ratio estimating system. In this system, a model representative of a behavior of the exhaust system is established. A single air-fuel ratio sensor is disposed in a confluent portion of the exhaust gas and outputs detected signals. The detected signals are inputted into the model to estimate the air-fuel ratio of each cylinder by means of an observer. A fuel injection quantity to each cylinder is corrected according to a deviation between an estimated air-fuel ratio and a target air-fuel ratio, so that the air-fuel ratio of each cylinder is brought to be consistent with the target air-fuel ratio. Besides, a time delay exists between a time when the exhaust gas flows into a vicinity of the air-fuel ratio sensor and a time when the air-fuel ratio is detected. This time delay is referred to as a response delay of exhaust system hereinafter. Considering that the response delay of exhaust system is varied due to a deterioration of responsiveness of the air-fuel ratio sensor, a deterioration parameter representative of the deterioration degree of the air-fuel ratio sensor is calculated by measuring a response delay of the air-fuel ratio sensor at the time of fuel-cut, and then sample timing (detecting timing) of the air-fuel ratio sensor is corrected based on the deterioration parameter.
However, in this system, when the air-fuel ratio is largely varied due to a disturbance while the response delay of the air-fuel ratio sensor is measured during the fuel-cut, the calculation accuracy of the deterioration parameter is deteriorated, so that the correction accuracy of the air-fuel ratio detecting timing is deteriorated.
JP-2001-140685A shows that when the air-fuel ratio is largely varied due to a disturbance while the response delay of the air-fuel ratio sensor is measured, the measurement of the response delay of the air-fuel ratio sensor is prohibited to avoid a deterioration of the correction accuracy of air-fuel ratio detecting timing.
A behavior at the time when cylinder-by-cylinder air fuel ratio disperses is a combination of the response delay of the exhaust system of each cylinder and the response delay of the air-fuel ratio sensor, which is repeated every 180° CA (in four cylinder engine). Hence, it is not always that the deviation of the air-fuel ratio detecting timing accurately agrees with the deviation of response delay of the air-fuel ratio sensor at the time of fuel-cut. Besides, a time period for correcting the deviation of the air-fuel ratio detecting timing is shorter than a time period of the response delay of the air-fuel ratio sensor. Thus, in the systems shown in JP-10-73049A (U.S. Pat. No. 5,806,506) and JP-2001-140685A, it is hard to accurately correct the deviation of the air-fuel ratio detecting timing.